Multivariable Partial Derivative
Input a function $f(x, y, z, …)$ and the variables, and I’ll calculate the partial derivatives.
Use standard notation: `*` for multiplication, `^` for exponents, `sin`, `cos`, `exp`, `log`, etc.
Results
Enter your function and variables above to see the results.
Understanding multivariable derivatives can be challenging, especially when dealing with functions of two or more variables. The Multivariable Derivative Calculator simplifies this process by automatically computing partial derivatives, gradients, and higher-order derivatives for complex multivariable functions — in just seconds.
Whether you’re a student learning calculus, a data scientist working with optimization problems, or an engineer dealing with physics equations, this calculator helps you differentiate multivariable functions easily and accurately.
🧠 What Is a Multivariable Derivative?
In calculus, a multivariable derivative measures how a function changes as one or more variables change. When a function depends on more than one variable (e.g., f(x, y, z)), you can find how it changes with respect to each variable individually — these are called partial derivatives.
For example:
If f(x, y) = x²y + 3xy², then:
- ∂f/∂x = 2xy + 3y²
- ∂f/∂y = x² + 6xy
These derivatives tell us how the function changes along each direction of the coordinate plane.
💡 Why Use a Multivariable Derivative Calculator?
This calculator is designed to handle complex differentiation tasks like:
- ✅ Partial Derivatives — Differentiate with respect to one variable while holding others constant.
- ✅ Gradient Vectors — Compute ∇f(x, y, z) to find the direction of steepest ascent.
- ✅ Higher-Order Derivatives — Find second or mixed partial derivatives such as ∂²f/∂x² or ∂²f/∂x∂y.
- ✅ Jacobian & Hessian Matrices — Evaluate multivariable functions in optimization and machine learning.
- ✅ Symbolic & Numeric Results — Get both exact expressions and decimal approximations.
It’s ideal for students, mathematicians, physicists, and professionals who need reliable, step-by-step multivariable differentiation.
🧮 How to Use the Multivariable Derivative Calculator
Using the calculator is simple and efficient. Here’s how:
- Enter your multivariable function.
Example:f(x, y, z) = x²y + sin(xy) + z³ - Select the variable(s) you want to differentiate with respect to (e.g., x, y, or z).
- Choose the derivative order.
- 1st order: ∂f/∂x
- 2nd order: ∂²f/∂x², ∂²f/∂x∂y
- 3rd order or higher if needed.
- Click “Calculate.”
The calculator instantly provides the derivative expression and (if requested) the step-by-step solution.
✍️ Example Calculations
Example 1: Partial Derivative
Given f(x, y) = 5x²y³ + 2xy, find ∂f/∂x. ∂f/∂x=10xy3+2y∂f/∂x = 10xy³ + 2y∂f/∂x=10xy3+2y
Example 2: Mixed Partial Derivative
Given f(x, y) = x²y + 4xy², find ∂²f/∂x∂y. ∂2f/∂x∂y=2x+8y∂²f/∂x∂y = 2x + 8y∂2f/∂x∂y=2x+8y
Example 3: Gradient Vector
If f(x, y, z) = x² + y² + z², then the gradient is: ∇f=(2x,2y,2z)∇f = (2x, 2y, 2z)∇f=(2x,2y,2z)
This vector points in the direction of the function’s steepest increase.
🧾 What the Calculator Displays
When you input a multivariable function, the calculator shows:
| Output Type | Description |
|---|---|
| Partial Derivative | Derivative of f with respect to a single variable |
| Mixed Derivative | Derivative of f with respect to multiple variables |
| Gradient Vector | Vector of all first-order partial derivatives |
| Hessian Matrix | Matrix of all second-order partials (optional) |
| Step-by-Step Work | Simplified process showing differentiation rules used |
🧩 Applications of Multivariable Derivatives
Multivariable calculus is foundational in many disciplines. Here’s where this calculator becomes invaluable:
1. Optimization Problems
Find minima or maxima of multivariable functions using partial derivatives and the gradient.
2. Machine Learning
Used in backpropagation and gradient descent algorithms for optimizing models.
3. Physics and Engineering
Calculate rate of change in fields like thermodynamics, electromagnetism, and fluid dynamics.
4. Economics
Determine marginal cost, revenue, or profit functions depending on several factors.
5. Statistics
Used in likelihood estimation and regression analysis.
🧭 Benefits of Using the Multivariable Derivative Calculator
- ⚙️ Instant results: Save time on lengthy differentiation steps.
- 📚 Step-by-step explanations: Understand every rule used.
- 🎯 Accurate computation: Ideal for checking homework or research work.
- 📈 Handles complex equations: Works with trigonometric, exponential, and logarithmic functions.
- 🧮 Flexible for any order: Supports 1st, 2nd, and higher-order derivatives.
🔍 Common Multivariable Derivative Rules
| Rule | Description | Formula |
|---|---|---|
| Power Rule | d/dx(xⁿ) = n·xⁿ⁻¹ | ∂/∂x(x²y³) = 2xy³ |
| Product Rule | ∂/∂x(uv) = u∂v/∂x + v∂u/∂x | ∂/∂x(x·y²) = y² + 2xy∂y/∂x |
| Chain Rule | ∂f/∂x = ∂f/∂u · ∂u/∂x | f(x, y(x)) |
| Sum Rule | ∂/∂x(f + g) = ∂f/∂x + ∂g/∂x | – |
| Mixed Partial Rule | ∂²f/∂x∂y = ∂²f/∂y∂x (if continuous) | – |
These are the foundational rules the calculator uses to generate accurate symbolic derivatives.
🧠 FAQs – Multivariable Derivative Calculator
1. What is a multivariable derivative?
It measures how a function changes with respect to one or more variables.
2. What is a partial derivative?
It’s the derivative of a multivariable function with respect to one variable, keeping others constant.
3. What is a mixed partial derivative?
It’s the derivative taken sequentially with respect to two or more variables (e.g., ∂²f/∂x∂y).
4. What is a gradient vector?
A vector containing all first-order partial derivatives — represents the direction of steepest increase.
5. What is a Hessian matrix?
A square matrix of second-order partial derivatives — used in optimization.
6. Can the calculator handle trigonometric functions?
Yes! It supports sin, cos, tan, exponential, and logarithmic functions.
7. What are higher-order partial derivatives?
They’re derivatives of derivatives — e.g., ∂²f/∂x², ∂³f/∂x³.
8. Is this calculator suitable for vector-valued functions?
Yes, it can compute Jacobians for multivariable vector functions.
9. How accurate is the result?
The calculator provides exact symbolic differentiation using algebraic rules.
10. Can I calculate directional derivatives?
Yes, by using the gradient vector and the direction vector.
11. What is the chain rule for multivariable functions?
It relates derivatives of composite functions with several variables.
12. Can it compute implicit derivatives?
Yes, if the function is given in implicit form.
13. What’s the difference between partial and total derivative?
Partial keeps other variables constant; total includes all variable dependencies.
14. What’s the use of the gradient in machine learning?
It’s used in gradient descent to minimize cost functions.
15. What is ∂²f/∂x∂y = ∂²f/∂y∂x called?
Clairaut’s Theorem on the equality of mixed partials.
16. Can I input more than three variables?
Yes, you can use as many as needed (x₁, x₂, x₃, etc.).
17. Does it show simplification steps?
Yes, the calculator simplifies expressions step by step.
18. Is it helpful for engineering students?
Absolutely — used in mechanics, thermodynamics, and electromagnetism.
19. Can I evaluate at specific points?
Yes, after differentiation, you can substitute numerical values.
20. Is it free to use?
Yes, the Multivariable Derivative Calculator is completely free and online.